For a partition $\lambda$ let $\chi_\lambda$ be the corresponding irreducible representation of the symmetric group $S_n$. Let $\mathrm{Imm}_\lambda(x) = \sum\limits_{\pi \in S_n} \chi_\lambda(\pi) x_{1 \pi(1)} \dotsm x_{n \pi(n)}$ be the immanant corresponding to $\lambda$.
Question 1: Is $\mathrm{Imm}_\lambda$ irreducible over $\mathbb{Z}$? If not, in how many irreducible factors does it split depending on $\lambda$?
It seems for all partitions of $n$ for $n$ at most 5 those polynomials are irreducible.
Question 2: Let $R=\mathbb{Q}(x_{i,j})/I$ where $I$ is the ideal generated by the $\mathrm{Imm}_\lambda$. What is the Krull dimension of $R$? The sequence starts with $2,6,12$ for $n=2,3,4$.